If the point (x, y) be equidistant from the points (a + b, b - a) and (a - b, a + b), prove that

Given points are A(a + b, b - a) and B(a - b, a + b). It is told that S(x, y) is equidistant from A and B.

So, we get SA = SB,

We know that distance between two points (x₁, y₁) and (x₂, y₂) is .

Now,

⇒ SA = SB

⇒ SA² = SB²

⇒ (x - (a + b))² + (y - (b - a))² = (x - (a - b))² + (y - (a + b))²

⇒ x² - 2(a + b)x + (a + b)² + y² - 2(b - a)y + (b - a)² = x² - 2(a - b)x + (a - b)² + y² - 2(a + b)y + (a + b)²

⇒ x(- 2a - 2b + 2a - 2b) = y(2b - 2a - 2a - 2b)

⇒ x(- 4b) = y(- 4a)

⇒ x(b) = y(a)

⇒

Applying componendo and dividendo,

⇒

∴ Thus proved.